A Riemannian Approach for Spatiotemporal Analysis and Generation of 4D Tree-shaped Structures

TL;DR

A novel mathematical representation of the shape space of 4D tree-shaped trajectories, a Riemannian metric on that space, and computational tools for fast and accurate spatiotemporal registration and geodesics computation between 4D tree-shaped structures are proposed.

Abstract

We propose the first comprehensive approach for modeling and analyzing the spatiotemporal shape variability in tree-like 4D objects, i.e., 3D objects whose shapes bend, stretch, and change in their branching structure over time as they deform, grow, and interact with their environment. Our key contribution is the representation of tree-like 3D shapes using Square Root Velocity Function Trees (SRVFT). By solving the spatial registration in the SRVFT space, which is equipped with an L2 metric, 4D tree-shaped structures become time-parameterized trajectories in this space. This reduces the problem of modeling and analyzing 4D tree-like shapes to that of modeling and analyzing elastic trajectories in the SRVFT space, where elasticity refers to time warping. In this paper, we propose a novel mathematical representation of the shape space of such trajectories, a Riemannian metric on that space, and computational tools for fast and accurate spatiotemporal registration and geodesics computation between 4D tree-shaped structures. Leveraging these building blocks, we develop a full framework for modelling the spatiotemporal variability using statistical models and generating novel 4D tree-like structures from a set of exemplars. We demonstrate and validate the proposed framework using real 4D plant data.

Figures (6)

• Figure 1: The 4D skeleton of Fig. 1 in the Supplementary Material, structured it into layers of branches. The red branch is the main branch, blue corresponds to the $2^{nd}$ layer of branches, yellow to the $3^{rd}$ layer, and purple to the $4^{th}$ layer.
• Figure 2: The proposed spatial registration. We also show four 3D tree shapes before and after their spatial registration using the proposed framework. Our spatial registration took on average $235$s to align two tomato and $17$s to align two maize 4D plants.
• Figure 3: The proposed framework for the analysis of 4D tree-shaped structures. The key idea is to represent 4D trees $H_1$ and $H_2$ as curves in the SRVFT space $\mathcal{C}_Q$, which is Euclidean but of infinite dimension. By learning a PCA subspace $\mathcal{C}_{\text{PCA}}$, 4D tree-shaped structures become curves in $\mathbb{R}^k$. However, instead of using the nonlinear elastic metric in $\mathcal{C}_T$ to model temporal variability, we further map the curves to the SRVF space where the $\mathbb{L}^2$ metric is equivalent to the full elastic metric. All the analysis can be performed in this space using standard vector calculus and mapped back to the original space for visualization. The computation time of the proposed temporal registration between two 4D plants is on average $0.037$s for tomato and $0.006$s for maize 4D plants.
• Figure 4: The 4D geodesic between the two 4D tree shapes of Fig. 8 in the Supplementary Material after spatiotemporal registration. The highlighted row is the mean 4D tree. This geodesic computation requires on average $0.025$s for 4D tomato plants.
• Figure 5: The mean 4D plant shape of the seven registered 4D tomato plants in Fig. 10-b in the Supplementary Material. The computation time is in the order of $0.0006$s.